OFFSET
1,2
COMMENTS
This sequence and A005843 and A130568 partition the positive integers into sequences with slopes t = 1+sqrt(5), u = 3+sqrt(5), v = 2, where 1/t + 1/u + 1/v = 1. The positions of 1 in A284620 are given by A005843, and of 2, by A130568.
From Michel Dekking, Mar 17 2020: (Start)
This sequence is a generalized Beatty sequence.
It was shown in the Comments of A284620 that A284620 is the letter-to-letter image of the fixed point x = ABCDABCDCD... of the morphism
mu: A->AB, B->CD, C->ABCD, D->CD,
with the letter-to-letter map lambda defined by
lambda: A->0, B->1, C->2, D->1.
Note that A284620(n)=0 iff x(n) = A, where x = ABCDABCDCD... is the fixed point of mu. The return words of A in x are ABCD and ABCDCD. Coding these two return words by their lengths, mu induces a morphism rho on the coded return words given by
rho(4) = 46, rho(4) = 466.
The difference sequence (a(n+1)-a(n)) equals the unique fixed point r = 4646646466... of rho.
The morphism g on the alphabet {a,b} given by
g(a) = ab, g(b) =abb
was introduced in A284620. We see that rho is just an alphabet change of the morphism g.
Let f given by f(b) = ba, f(a) = b be the Fibonacci morphism on the alphabet {b,a} with fixed point x_F = babbababba....
Let x_G = ababbababb... be the fixed point of g. It is well-known (see, e.g., Lemma 12 in "Morphic words..."), that x_G = a x_F.
In general the partial sums of x_F are equal to the generalized Beatty sequence V given by V(n) = p*floor(n*phi) +q*n+r, where p = a-b and q = 2*b-a. See Lemma 8 in the Allouche and Dekking paper. Here we obtain p = 2, q = 2. So a(n) = 2*floor((n-1)*phi) + 2*n - 1, for n>0.
(End)
LINKS
Clark Kimberling, Table of n, a(n) for n = 1..10000
J.-P. Allouche, F. M. Dekking, Generalized Beatty sequences and complementary triples, Moscow Journal of Combinatorics and Number Theory 8, 325-341 (2019).
M. Dekking, Morphic words, Beatty sequences and integer images of the Fibonacci language, Theoretical Computer Science 809, 407-417 (2020).
FORMULA
a(n+1) = 2*A001950(n) + 1, n>0. - Michel Dekking, Mar 17 2020
EXAMPLE
As a word, A284620 = 012101212101210121..., in which 0 is in positions 1,5,11,15,...
MATHEMATICA
s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0}}] &, {0}, 13] (* A003849 *)
w = StringJoin[Map[ToString, s]]
w1 = StringReplace[w, {"00" -> "2"}]
st = ToCharacterCode[w1] - 48 (* A284620 *)
Flatten[Position[st, 0]] (* A284621 *)
Flatten[Position[st, 1]] (* A005843 *)
Flatten[Position[st, 2]] (* A130568 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, May 02 2017
STATUS
approved